Wireless receivers in mobile devices rely upon mobile connections that consist of strictly regulated, narrow RF bands. Band allocation minimizes interference but also mandates that the entire commercial (Continental US) cellular traffic occupies less than a GHz. This provides high mobility, but one penalty for the unrestricted mobility of mobile devices is paid in terms of low connectivity: while a single wired rate now exceeds 10 Gbps, the entire cellular network is confined to only 765 MHz. Remarkably, the modern mobile network must accommodate all of its users within the spectral range that is orders of magnitude narrower than that of a single physical wireline.
The available radio frequency (RF) spectrum is scarce and is tightly regulated to avoid interference issues. High speed and reliable data transmission communications must be enabled while avoiding interference and adhering to strictly regulated spectral windows allocated for cellular, military, navigation and broadcast services. The strict band allocations minimize interference but less than a GHz wide band available for cellular data traffic. The spectral range allocated such traffic is four orders of magnitude narrower than that of a single physical wireline. Hilbert, M., & López, P., “The world's technological capacity to store, communicate, and compute information,” Science, 332(6025), 60-65 (2011). To transmit freely across the entire RF range, signal spectral power density (SPD) must be decreased below the band-specific interference threshold. A method to achieve arbitrarily low SPD is spectral spreading of the channel. Such spectral spreading makes signal reception and reconstruction a processing challenge, which is addressed with increased processing power in modern portable devices. See, e.g., Mitchell, T., “Broad is the way: ultra-wideband technology,” IEEE Review, 47(1), 35-39 (2001); Ferrain, I., et al., “Multigate transistors as the future of classical metal-oxide-semiconductor field-effect transistors,” Nature 479, 310-316 (2011). The normal approach is to digitize the ultrawideband signal and compute its correlation with the spreading sequence.
In a spread-spectrum channel, data is rapidly modulated by a specific sequence (codeword) to produce a waveform with much wider bandwidth. By spreading the signal over a wide bandwidth, the SPD in each frequency is greatly reduced. The spreading also reduces the effects of interference and can provide security, as well. Intuitively, SPD can be arbitrarily lowered by a mere increase in code rate, implying that a regulation-free transmission across the entire RF range can be attained. As an illustration, to reduce SPD of a MHz-wide channel by 10,000 times, its physical bandwidth should be increased to 10 GHz. While multi-GHz modulation is easily accomplished, the reconstruction (decoding) of 10 GHz-wide spread-spectrum channel poses a significant computational challenge. The received signal must be synchronized to within a fraction of 100 ps, sampled and quantized at the spreading rate (10 GHz) and finally correlated with the code. In addition to this decoding challenge, the interference generated by different arrival paths to the receiver must be processed out. Choi, J. D., & Stark, W. E., “Performance of ultra-wideband communications with suboptimal receivers in multipath channels,” Selected Areas in Communications, IEEE Journal on, 20(9), 1754-1766 (2002). These can be accomplished, at least in principle, by repeatedly computing a real-time Fourier transform of the received signal. Unfortunately, the last requirement calls for processors approaching 1012 floating-point operations per second (TFLOPS), well outside the mobile dissipation envelope. Jeon, D., et al, “Energy-optimized high performance FFT processor,” Acoustics, Speech and Signal Processing (ICASSP), 2011 IEEE International Conference on (pp. 1701-1704). IEEE. (2011, May); Tang, S. N., et al., “A 2.4-GS/s FFT processor for OFDM-based WPAN applications,” Circuits and Systems II: Express Briefs, IEEE Transactions on, 57(6), 451-455 (2010).
To transmit a signal in interference-free manner, normal spread-spectrum transmission broadens a signal bandwidth δf to Δf (where δf/<<Δf), reducing its spectral power density below band-regulated level ρ(f). Spectral broadening is achieved by imposing a unique codeword c(t), modulated at chip rate Δf. A transmission impaired, spectrally broadened signal must be sampled at the spreading rate (Δf) and subsequently decoded in real time. The decoding requires codeword synchronization with sub-chip precision and can be achieved in spectral domain by single multiplication.
When a channel experiences negligible multi-path interference (MPI), the encoded bit b[n] can be recovered by correlating the quantized received signal x[m] and the codeword c[m]. In Fourier domain, correlation is mapped to a single multiplication:
                              b          ⁡                      [            n            ]                          =                              ∑                          m              =                                                -                  N                                /                2                                                    N              /              2                                ⁢                                          ⁢                                    c              ⁡                              [                m                ]                                      ⁢                                          x                ⁡                                  [                                      m                    +                    n                                    ]                                            ⁢                              ⟶                𝒥                            ⁢                                                                    C                    ~                                    *                                ⁡                                  [                  k                  ]                                                      ⁢                                          X                ~                            ⁡                              [                k                ]                                                                        (        1        )            where {tilde over (C)} and {tilde over (X)} are the discrete Fourier transform (DFT) of the codeword and the received signal. In temporal domain, when the delay between the received signal and codeword is not known, the discrete summation in Eq. 1 must be performed repeatedly to recover data. However, this approach mandates for DFT of the received signal to be computed in real time, posing a progressively larger challenge as the spreading rate increases.